Overview

Thank you for volunteering to participate in our study. We believe that this task will take approximately 20 - 25 minutes.

In this document, you will read about the design and data collection of an experiment that was performed by Jansen et al. We will describe a Bayesian model which can be used to analyse the data for that experiment. Your goal is to set the priors probability distribution for each parameter in this model.

This study is divided into three parts:

  1. In the first part, we will provide you with:
    • the experiment design,
    • relevant information from a prior study which was performed using a similar set up,
    • the description and parameterisation of a simple regression model with two parameters,
    • an interactive visualization, for each parameter, to show the effect of prior distributions, using two different families. (Note: this part is to familiarise you with the visualizations).
  2. In the next part, you will be presented with the visualizations again. Here, you will submit your choice of priors
  3. Finally, we will ask you a set of open-ended questions about the visualization.

Experiment and data

Study goals

Jansen et al. performed an experiment to examine the effect of different incidental power poses on risk-taking behavior. They used the Balloon Analogous Risk Task (BART), which is a standard test in Psychology, to measure people’s risk-taking behavior in the form of a game. The task was administered through a digital interface.

Participants were placed in either the constrictive or expansive condition. The expansive and constrictive conditions were created by manipulating the location of the buttons on a table-top interface. In the constrictive condition the buttons were placed around a small (0.15m2); in the expansive condition, the buttons were placed around a large (0.6m2) interface (the Figure below shows what the posture looks like; buttons are not shown in this image).

Design of the BART experiment

The basic task in BART is to pump up a virtual balloon using on-screen buttons. With each pump, the balloon grows a bit and the player gains a point, which are linked to monetary rewards — the more the players pump up the balloons, the higher their payoff. The maximum size of a balloon is reached after 128 pumps. The risk is introduced through a random, uniformly distributed, point of explosion for each balloon with the average and median explosion point at 64 pumps. The optimal strategy to maximise payoff is to perform 64 pumps. Each participant repeats this 30 times.

Prior studies

BART tasks have been commonly used in psychology research to assess risk-taking behavior. A meta-analysis of 22 studies which used the BART task found that the average number of pumps (averaged across conditions) to vary between 24.60 to 44.10 (out of 128 total possible pumps), with a weighted standard deviation of 5.93. This means that based on prior studies, on average, participants in the BART task are most likely to be risk-averse.

Analysis

Regression Model

One way of analysing this data is to use a poisson regression model: the outcome variable will be the number of pumps by the participant, and the predictor will be a (categorical) dummy variable indicating which condition the participant is in.

\[ pumps_i \sim Poisson(\lambda_i) \\ log(\lambda_i) \sim \alpha + \beta \times condition_i \] where, \(condition_i\) has two levels: 0 for the constrictive condition, and 1 for the expansive condition.

Priors

Since this is a Bayesian analysis, we’ll need to specify prior distributions for \(\alpha\) and \(\beta\). The prior distributions indicate the probability mass that will be assigned to different values for that parameter, and express the best prior information, before seeing any data, of reasonable values for the model parameters.

Your goal is to identify the best possible priors for \(\alpha\) and \(\beta\).

You can choose priors from any reasonable family of distributions. In this study, we let you choose between:

  • Student-t priors
  • Normal priors

These families can be characterised using their mean, \(\mu\) and standard deviation, \(\sigma\). The values of \(\mu\) and \(\sigma\) will affect your result. In the interactive visualization, you can change the mean and standard deviation of the distribution, and see how this affects probability density of the parameter.

There are different ways to set a prior:

  1. use information from previous literature,
  2. use expert domain knowledge, or
  3. specify “weakly informative priors” that only minimally restrict the model, and excludes only values that are implausible or impossible.

Interactive visualizations

Student-t priors

Interactive visualization of the prior on the intercept parameter, \(\alpha \sim \mathrm{Normal(\mu_{\alpha}, \sigma_{\alpha})}\)

Interact with the visualization by dragging the black dot in the widget across the space of possible specifications of the prior distribution. This shows how the probability density of the prior changes as the parameters change.

Interactive visualization of the prior on the mean difference parameter, \(\beta \sim \mathrm{Normal(\mu_{\beta}, \sigma_{\beta})}\)

Normal priors

Interactive visualization of the prior on the intercept parameter, \(\alpha \sim \mathrm{Normal(\mu_{\alpha}, \sigma_{\alpha})}\)

Interactive visualization of the prior on the mean difference parameter, \(\beta \sim \mathrm{Normal(\mu_{\beta}, \sigma_{\beta})}\)

Now that you’ve interacted with the visualizations, you’ll need to choose one prior distribution for each parameter. Please proceed to the next page.